If the abundances of the constituent molecules of a biochemical reaction system are sufficiently high then their concentrations are typically modelled by a coupled set of ordinary differential equations (ODEs). If, however, the abundances are low then the standard deterministic models do not provide a good representation of the behaviour of the system and stochastic models are used. In this talk, I will first introduce both the stochastic and deterministic models. I will then provide theorems that allow us to determine the qualitative behaviour of the underlying mathematical models from easily checked properties of the associated reaction network. I will present results pertaining to so-called ``complex-balanced'' models and those satisfying ``absolute concentration robustness'' (ACR). In particular, I will show how ACR models, which are stable when modelled deterministically, necessarily undergo an extinction event in the stochastic setting. I will then characterise the behaviour of these models prior to extinction.
Tmoslav Plesa: Chemical Reaction Systems with a Homoclinic Bifurcation: An Inverse Problem, 25+5 min;
John Ockendon: Wave Homogenisation, 10 min + questions;
Hilary Ockendon: Sloshing, 10 min + questions
We discuss moments of $L$-functions in function fields, in the hyperelliptic ensemble, focusing on the fourth moment of quadratic Dirichlet $L$-functions at the critical point. We explain how to obtain an asymptotic formula with some of the secondary main terms.